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ON THE MOTION OF  PENDULUMS.                         129
which in fact is the same as the second of the equations (8).    The solution thus obtained is as we have seen
/ denoting a function the form of which there is no need to write down, which satisfies (180) when written for v'. Now it will be seen at once that the expression (181) satisfies the exact equation (179), and therefore the approximate solution obtained by the method of Art. 8 is in fact exact, except so far as regards the termination of the disk at its edge, which is what it was required to prove.
Passing from semi-polar to polar co-ordinates, by putting 2 = rcos#, -sr = rsin0, we get from (179), after writing pp for P>
2 dv'         1      d     .      di/\         v      _ I dv'
~      dt
Suppose now the solid to be a sphere, having its centre at the origin. Let a be its radius, y its angular velocity, and suppose the fluid initially at rest. Then v' is to be determined from the general equation (182) and the equations of condition
v' = Q when t = 0,     v = a y sin 0 when r = a,     vf = 0 when r = oo . All these equations are satisfied by supposing
t/ = fl"sin0,
v" being a function of r and t only.    We get from (182) <W    2dv"    2v"     I civ"
If we suppose # constant, v" will tend indefinitely to become
fjfjj constant as t increases indefinitely, and in the limit ~=- = 0, whence
Cut
we get from (183) and the equations of condition v" = ax when r = a, v" = 0 when r = oo ,
,          -.
r                 r
This is the solution alluded to in Art. 8 of my paper On the Theories of the Internal Friction of Fluids in motion, <&c.
s. in.